The 19-year-old Millwall striker scored twice before the break, firing his side ahead after 10 minutes and then increasing their advantage eight minutes before half-time.
Wolves midfielder Anthony Forde completed the job when he converted a 57th-minute penalty to cap a second successive victory for Noel King's side, who ended the qualifying campaign for the 2013 European Championships with a 4-2 win in Italy in September last year.
The Republic got off to a shaky start, with keeper Aaron McCarey having to keep out a miscued third-minute header by team-mate Derrick Williams before Dutch skipper Alex Schalk hit the post with a deflected shot.
But O'Brien set them on their way when he tapped home from Forde's cross after it eluded the entire Netherlands defence.
The teenager might have claimed a second with 22 minutes gone, but dragged his shot wide.
However, he did not have to wait long to extend Ireland's lead, turning home Everton defender Shane Duffy's flick-on from a Forde free-kick.
O'Brien hobbled off 11 minutes after the restart after falling awkwardly as he challenged for a high ball, but the game was effectively over seconds later when Forde was fouled by Stefano Denswil and picked himself up to convert the resulting penalty.
Substitute keeper Sean McDermott had to make an important save at the death to deny Joel Veltman a consolation goal but Ireland were worthy winners over a Dutch side who qualified for this summer's Euro 2013 finals as group winners.
The victory will give King's men a massive boost in confidence as they prepare for the new qualifying campaign, which gets underway in the Faroe Islands in August.